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	Constructivism (mathematics) - Wikipedia, the free encyclopedia
		Constructivist mathematics uses constructivist logic, which is essentially a removal of the law of the excluded middle from classical logic.
		In the constructive version, it is required that, for any given distance, it is possible to actually specify the point in the sequence where this happens (this required specification is often called the modulus of convergence).
		Constructions can be defined as broadly as free choice sequences, which is the intuitionistic view, or as narrowly as algorithms (or more technically, the recursive functions), or even left unspecified.
	en.wikipedia.org /wiki/Mathematical_constructivism (973 words)

	Constructive Mathematics
		Constructive mathematicians believe that numbers, not sets, are the basis of mathematics and that since infinite sets do not seem to exist in nature, mathematicians can have no real, intuitive understanding on which to base judgments concerning them.
		Constructive mathematicians believe every problem in mathematics can be reduced to a numerical problem such as Goldbach's conjecture and every numerical question can be reduced to a problem of deciding whether or not a 1 appears in a sequence of O's and 1's.
		In Bishop's constructive revision of set theory, however, the axiom of choice appears to be noncontentious, because it is stipulated that to call a set nonempty one must show that the construction method specified in the definition of the set actually generates a member of the set.
	www.digitalphysics.org /Publications/Cal79/html/cmath.htm (8036 words)

	Constructive Mathematics
		It is important to realize that constructive algebra is algebra; in fact it is a generalization of classical algebra in that we do not assume the law of excluded middle, just as group theory is a generalization of abelian group theory in that the commutative law is not assumed.
		A constructive proof of a theorem is, in particular, a proof of that theorem.
		Ruitenburg, A course in constructive algebra, Springer-Verlag 1988.
	www.math.fau.edu /Richman/HTML/CONSTRUC.HTM (1293 words)

	Beginners’ References - Constructive Mathematics - Mathematics and Statistics Department - University of ... (Site not responding. Last check: 2007-10-09)
		Constructive mathematics is distinguished from its traditional counterpart, classical mathematics, by the strict interpretation of the phrase “there exists” as “we can construct”.
		First, it leads to a form of constructive mathematics, in which large parts of classical mathematics are rejected.
		Second, the reliance on a philosophy of mind introduces features that are absent from classical mathematics as well as from other forms of constructive mathematics: unlike those, intuitionistic mathematics is not a proper part of classical mathematics...
	pent-9.math.canterbury.ac.nz /php/groups/cm/references (473 words)

	Constructive Mathematics
		Constructive mathematics is based on the idea that the logical connectives and the existential quantifier are to be interpreted as instructions on how to construct a proof of the statement involving these logical expressions.
		Unfortunately--and perhaps inevitably, in the face of opposition from mathematicians of such stature as Hilbert--Brouwer's intuitionist school of mathematics and philosophy became more and more involved in what, at least to classical mathematicians, appeared to be quasi-mystical speculation about the nature of constructive thought, to the detriment of the practice of constructive mathematics itself.
		Taking the logic as the characteristic of constructive mathematics, it does not, of course, reflect the primacy of mathematics over logic that was part of the belief of Brouwer, Heyting, Markov, Bishop, and other pioneers of constructivism.
	www.seop.leeds.ac.uk /archives/fall1998/entries/mathematics-constructive (1652 words)

	Constructive Mathematics
		Constructive mathematics is pertinent to the PRL project because all of our logics have had a constructive core.
		Prominent points have been the Greek interest in ruler and compass constructions, the Arab interest in algorithms and the algebra of solving equations, and the European development of the eighteenth and nineteenth century theory of functions.
		Judith Underwood gave a constructive proof of the completeness of Intuitionistic Propositional Calculus with respect to Kripke models in a paper and her thesis.
	www.cs.cornell.edu /Info/Projects/NuPrl/Intro/ConstrMath/constrmath.html (917 words)

	WebGuest Keyboard-Free Directory - Category: Society : Philosophy : Philosophy of Science : Mathematics (Site not responding. Last check: 2007-10-09)
		Constructive Mathematics - Constructive mathematics is distinguished from its traditional counterpart, classical mathematics, by the strict interpretation of the phrase `there exists' as `we can construct'.
		Inconsistent Mathematics - Inconsistent mathematics is the study of the mathematical theories that result when classical mathematical axioms are asserted within the framework of a (non-classical) logic which can tolerate the presence of a contradiction without turning every sentence into a theorem.
		Mathematical Structures Group - Research topics include mathematical models and theories in the empirical sciences, models and theories in mathematics, category theory, and the use of mathematical structures in theoretical computer science.
	www.webguest.com /Society/Philosophy/Philosophy_of_Science/Mathematics (540 words)

	Constructive Mathematics
		Mathematics arises when the subject of two-ness, which results from the passage of time, is abstracted from all special occurrences.
		Normally, a sequence in constructive mathematics is given by a rule which determines, in advance, how to construct each of its terms; such a sequence may be said to be lawlike or predeterminate.
		However, the comparison with classical mathematics should not be made superficially: in order to understand that there is no real contradiction here, we must appreciate that the meaning of such terms as “function” and even “real number” in intuitionistic mathematics is quite different from that in the classical setting.
	plato.stanford.edu /entries/mathematics-constructive (6375 words)

	Mathematical constructivism : Constructive mathematics (Site not responding. Last check: 2007-10-09)
		In the philosophy of mathematics, mathematical constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists.
		When you assume that an object doesn't exist, and derive a contradiction from that assumption, you still have not found it, and therefore not proved its existence, according to constructivists.
		Constructivism is often confused with mathematical intuitionism, but in fact, intuitionism is only one kind of constructivism.
	www.termsdefined.net /co/constructive-mathematics.html (284 words)

	Douglas S. Bridges
		`Constructive mathematics and unbounded operators- a reply to Hellman', J. Philosophical Logic 24, 549- 561, 1995.
		`A constructive analysis of a proof of that the numerical range is convex' (with Robin Havea), London Math.
		`Constructive unbounded operators' (with Hajime Ishihara), in: Reuniting the Antipodes- Constructive and Nonstandard Views of the Continuum (Berger, Ulrich, Horst Osswald, and Peter Schuster, editors), 25- 37, Proceedings of the Symposion in San Servolo/Venice, Italy, May 17-22, 1999, Synthese Library, Kluwer Academic Publishers, Dordrecht, December 2001.
	www.math.canterbury.ac.nz /~mathdsb/complete.html (1915 words)

	Constructive mathematics
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	www.geocities.com /mafika/constructive_mathematics.html (340 words)

	Mathematics - Directory
		Because these principles also underly Russian recursive analysis and the constructive analysis of E. Bishop and his followers, intuitionistic logic may be considered the logical basis of constructive mathematics.
		Inconsistent mathematics is the study of the mathematical theories that result when classical mathematical axioms are asserted within the framework of a (non-classical) logic which can tolerate the presence of a contradiction without turning every sentence into a theorem.
		Research topics include mathematical models and theories in the empirical sciences, models and theories in mathematics, category theory, and the use of mathematical structures in theoretical computer science.
	society.codpex.com /Philosophy/Philosophy_of_Science/Mathematics (457 words)

	Citations: Foundations of Constructive Mathematics - Beeson (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		A Constructive Formalization of the Fundamental Theorem of..
		This is achieved through dividing the interval [a; b] into n subintervals of length (appealing to the archimedian axiom) then, using a tricky induction argument and the properties of the less than relation, we nd an i such that a i y a i 2, from which it is easy to prove that....
		An example is the construction that introduces the reals as in nite sequences of digits [PEE97, Wei00, CDG00] in this case the equivalence follows from the fact that it is possible to transform....
	citeseer.lcs.mit.edu /context/2835/0 (4176 words)

	Citations: Constructive mathematics and computer programming - of (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		A type system is constructed by selecting certain terms of the computation system to denote types and by specifying for each type what terms are its members and when two members of the type are to be considered equal.
		A meta mathematical investigation of Martin L of type theory is carried out in (Aczel, 1980) Unfortunately, the perspectives of model theoretic semantics and of Martin L of type theory are at odds.
		A meta mathematical investigation of Martin Lof type theory is carried out in [1] Unfortunately, the perspectives of model theoretic semantics and of Martin Lof type theory are at odds.
	citeseer.lcs.mit.edu /context/327/0 (6135 words)

	Read This: Essays in Constructive Mathematics
		Although admittedly part of the problem was my students' lack of mathematical maturity, I had to agree with them that the author's explanations were often cryptic, and the point of some of the exercises obscure, unless one was already aware of the historical and mathematical issues at stake.
		Bonnie Shulman is associate professor of mathematics at Bates College in Lewiston, ME. Coming of age in the 1960s, she has always been attracted to theories and viewpoints that challenge the status quo.
		Trained as a mathematical physicist, her current intellectual passions include history and philosophy of mathematics and applications of game theory to studying social conflict and cooperation.
	www.maa.org /reviews/constructiveessays.html (1360 words)

	IBFI Schloss Dagstuhl - Dagstuhl Seminar 05021
		The goal of the seminar is to bring together people from the communities of formal proofs, constructive mathematics and computer algebra (in a wide meaning).
		One objective of the seminar is to bridge the gap between conceptual (abstract) and computational (constructive) mathematics, by providing a computational understanding of abstract mathematics.
		It is remarkable that in constructive and computer algebra, progress in sophisticated algorithms often implies progress on basics.
	www.dagstuhl.de /05021 (785 words)

	Constructive Mathematics, in Theory and Programming Practice - Bridges, Reeves (ResearchIndex)
		It gives a sketch of both Myhill's axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focusses on the relation between constructive mathematics and programming, with emphasis on Martin-Lof's theory of types as a formal system for BISH.
		54 The formula--as--types notion of construction (context) - Howard - 1980
		3 The fundamental theorem of algebra: a constructive developme..
	citeseer.ist.psu.edu /319229.html (706 words)

	Amazon.ca: Books: Essays in Constructive Mathematics (Site not responding. Last check: 2007-10-09)
		This means that its definitions and proofs use finite algorithms, not `algorithms' that require surveying an infinite number of possibilities to determine whether a given condition is met.
		It is not surprising that the first two topics can be treated constructively---although the constructive treatments shed a surprising amount of light on them---but the last topic, involving integrals and differentials as it does, might seem to call for infinite processes.
		Readers of his Advanced Calculus will know that his preference for constructive mathematics is not new.
	www.amazon.ca /exec/obidos/ASIN/0387219781 (483 words)

	Special IDR Meeting (Site not responding. Last check: 2007-10-09)
		His contributions to the mathematical sciences and to society at large have resulted in his election to both the U.S. National Academy of Science and the National Academy of Engineering.
		He is also a fellow of the American Academy of Arts and Sciences, a member of the Academia Leopoldina (Deutsche Akademie der Naturforscher), and a foreign member of the Polish Academy of Sciences.
		He is currently a Steenbock Professor of Mathematics and Computer Sciences at the University of Wisconsin - Madison.
	www.waveletidr.org /dagstuhl_meeting/dagstuhl.html (389 words)

	Conference on experimental and constructive mathematics
		Conference on experimental and constructive mathematics 97, SFU, June 23-24, 1997
		This is a collection of (informal) talks in and around the general areas of interest to CECM (analysis, computational mathematics, experimental mathematics, number theory, symbolic algebra, and more).
		It is sponsored by the Pacific Institute for the Mathematical Sciences (PIms) and the Centre for Experimental and Constructive Mathematics (CECM).
	www.pims.math.ca /science/1996/exthem96.12.html (277 words)

	A Catalog of Mathematics Resources on WWW and the Internet
		University of Cambridge, England - Faculty of Mathematics.
		Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada.
		Mathematics, Science and Technology Education - from the University of Illinois, Urbana-Champaign.
	mthwww.uwc.edu /wwwmahes/files/math01.htm (2091 words)

	Discussion. Applied constructive mathematics: on Hellman's 'mathematical constructivism in spacetime' -- Billinge 51 ...
		Applied constructive mathematics: on Hellman's 'mathematical constructivism in spacetime' -- Billinge 51 (2): 299 -- The British Journal for the Philosophy of Science
		Applied constructive mathematics: on Hellman's 'mathematical constructivism in spacetime'
		Hellman ([1998]) claims that constructive mathematics is inadequate
	bjps.oupjournals.org /cgi/content/abstract/51/2/299 (194 words)

	CECM - Centre for Experimental and Constructive Mathematics (Site not responding. Last check: 2007-10-09)
		The Centre for Experimental and Constructive Mathematics (CECM), a Simon Fraser University research centre within the Departments of Mathematics and Statistics and Actuarial Science, is located at SFU's Burnaby Campus in British Columbia (Canada).
		CECM's mandate is to explore and promote the interplay of conventional mathematics with modern computation and communication in the mathematical sciences.
		This honour was awarded jointly by the Mathematical Programming Society and the Society for Industrial and Applied Mathematics at the 18th International Symposium on Mathematical Programming, Copenhagen, August 18, 2003.
	cgi.cecm.sfu.ca (301 words)

	Other Mathematics Service Providers
		University of Calgary Department of Mathematics and Statistics
		University of New Brunswick Department of Mathematics and Statistics
		Mathematics Departments in the United States (listed alphabetically and geographically)
	camel.math.ca /Services/web.html (469 words)

	Non-Constructive Computational Mathematics (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		The implementation does not mention the ¯ operator explicitly, but has the strength to define the ¯ operator through the built-in...
		111 Constructivism in Mathematics (context) - Troelstra - 1988
		25 A Mathematical Incompleteness in Peano Arithmetic (context) - Paris, Harrington - 1978
	citeseer.ist.psu.edu /kunen95nonconstructive.html (351 words)

	CSS - Groups - Centre for Experimental and Constructive Mathematics
		CECM's mandate is to explore and promote the interplay of conventional mathematics and modern computation.
		The Centre provides a sophisticated but easy to use computational environment for mathematical research.
		CECM's resources are available to all its members and, where possible, to other interested individuals both at Simon Fraser and elsewhere via associate memberships and the World Wide Web.
	css.sfu.ca /groups/group.cgi?GroupID=18 (76 words)

	Cogprints - A Unifying Field in Logics: Neutrosophic Logic. (Site not responding. Last check: 2007-10-09)
		Session of the ''Mathematical Foundation of Artificial Intelligence'' Seminar, Sofia, October 5, 1994, Preprint MRL-MFAIS-10-94, Sofia, 1994, 1-13.
		Session of the ''Mathematical Foundation of Artificial Intelligence'' Seminar, Sofia, October 5, 1994, Preprint MRL-MFAIS-10-94, Sofia, 1994, 14-26.
		Session of the ''Mathematical Foundation of Artificial Intelligence'' Seminar, Sofia, October 5, 1994, Preprint MRL-MFAIS-10-94, Sofia, 1994, 27-35.
	cogprints.org /1919 (2954 words)

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